Question

please solve this class 9 question ​

Answer 1

$a = \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } \: \: and \: b = \frac{ \sqrt{5 } - \sqrt{2} }{ \sqrt{5} + \sqrt{2} }$

$we \: have \: to \: find \: \frac{ {a}^{2} + ab + {b}^{2} }{ {a}^{2} - ab - {b}^{2} } \: = \frac{( {a + b})^{2} }{({a - b})^{2} }$

$a + b = \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } + \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} } \\ a + b = \frac{ ({ \sqrt{5} + \sqrt{2} })^{2} + ({ \sqrt{5} - \sqrt{2} })^{2} }{( \sqrt{5 } - \sqrt{2} )( \sqrt{5} + \sqrt{2} )} = \frac{14}{3}$

$a - b \: = \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } - \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} } \\ a - b \: = \frac{ { (\sqrt{5} + \sqrt{2} )}^{2} - {( \sqrt{5} - \sqrt{2} )}^{2} }{( \sqrt{5} + \sqrt{2} )( \sqrt{5} - \sqrt{2}) } \\ a - b = \frac{2 \sqrt{10} }{3} \\ ( {a - b})^{2} = \frac{40}{9} \\( {a + b})^{2} = \frac{196}{9}$

$\frac{ ({a + b})^{2} }{( {a - b})^{2} } = \frac{196}{9} \times \frac{9}{40} \\ = \frac{49}{10} = 4.9$